Modeling #1 - Questions and Answers

Question 1 To describe: The difference between (fg)(x) and (f · g)(x) Answer 1 Formula used: Product of two functions: (fg)(x) = f(x)g(x) Composition of two functions: (f ∘ g)(x) = f(g(x)) Calculation: The function (fg)(x) denotes the product of two functions f and g and can be written as (fg)(x) = f(x)g(x). The function (f ∘ g)(x) denotes the composition of two functions f and g and can be written as (f ∘ g)(x) = f(g(x)). Let us see their examples. Suppose two functions are: f(x) = 3x², g(x) = x + 1 Therefore, (fg)(x) = f(x)g(x) = 3x²(x + 1) = 3x³ + 3x² (f ∘ g)(x) = f(g(x)) = 3(g(x))² = 3(x + 1)² = 3(x² + 2x + 1) = 3x² + 6x + 3 Conclusion: The function (fg)(x) denotes the product of two functions f and g and can be written as (fg)(x) = f(x)g(x). The function (f ∘ g)(x) denotes the composition of two functions f and g and can be written as (f ∘ g)(x) = f(g(x)). Answer 2 Step 1: Given: (f ∘ g)(x) represents the composition of two functions, f and g. It means that we first apply the function g to the input x, and then we apply the function f to the result. Mathematically, (f ∘ g)(x) can be written as f(g(x)). On the other hand, (f* g)(x) represents the pointwise multiplication or product of two functions, f and g. It means that for each value of x, we multiply the outputs of f(x) and g(x). Mathematically, (f* g)(x) can be written as f(x) * g(x). Step 2: Finally, the difference between (f ∘ g)(x) and (fg)(x) lies in the operations performed. (f ∘ g)(x) represents the composition of two functions, where we apply g first and then f, while (fg)(x) represents the pointwise multiplication of two functions, where we multiply the outputs of f(x) and g(x) for each input value of x. Here's a visual representation of the difference: For (f ∘ g)(x): Apply g to x: g(x) Apply f to the result: f(g(x)) For (f*g)(x): Evaluate f(x): f(x) Evaluate g(x): g(x) Multiply the results: f(x) * g(x) Answer 3 To describe the difference between (fg)(x) and (f * g)(x): (fg)(x) represents the composition of two functions f and g, where the output of g is used as the input for f. In mathematical notation, it can be written as (fg)(x) = f(g(x)). On the other hand, (fg)(x) represents the product of two functions f and g, where the output of f is multiplied by the output of g. In mathematical notation, it can be written as (fg)(x) = f(x) * g(x). In summary, (fg)(x) is the composition of functions, while (f*g)(x) is the product of functions. Answer 4 Answer: (fg)(x) is the product of f(x) and g(x): (fg)(x) = f(x) * g(x). (f* g)(x) is the convolution of f(x) and g(x): (fg)(x) = ∫ f(t)g(x - t) dt. Explanation: In mathematics, (fg)(x) represents the product of two functions, f(x) and g(x). It is obtained by multiplying the outputs of f(x) and g(x) evaluated at the same input, x. Mathematically, it can be expressed as (fg)(x) = f(x)g(x). On the other hand, (fg)(x) represents the convolution of two functions, f(x) and g(x). Convolution is an operation that combines two functions to produce a third function, which describes how the shape of one function affects the other as it is shifted and scaled. Mathematically, the convolution (f g)(x) is defined as: (fg)(x) = ∫ f(t)g(x - t) dt In simpler terms, to evaluate (f g)(x), we take one function, f(x), and "slide" it across the other function, g(x), while multiplying their values at each point and adding up the results. Therefore, the main difference between (fg)(x) and (f* g)(x) lies in the operation performed. (fg)(x) represents a point-wise multiplication of two functions, while (f*g)(x) represents their convolution. Question 2 Which of the following are true about regression with one predictor variable (often called simple regression)? Check all that apply. a) The regression equation is the line that best fits a set of data as determined by having the least squared error. b) The slope describes the amount of change in Y for a one-unit increase in X. c) After conducting a hypothesis test that the slope of the regression equation is nonzero, you can conclude that your predictor variable, X, causes Y. Answer Step 1 Here option A is correct as it is the definition of the regression equation. Here option B is also correct as slope is the change in Y by one unit increase in X. Here option C is not correct as association or correlation doesn't mean causation. Question 3 To find f(x + h) and the difference quotient of f and simplify. Given: f(x) = 1 − x³ Answer Formula used: a) Identity: (a + b)³ = a³+3a²b+3ab²+b³ b) To find f(x + h) we shall plug x + h in the function c) Difference quotient = (f(x+h) - f(x)) / h Calculations: a) We have f(x) = 1 - x³. To find f(x + h) we plug (x + h) in place of x. So f(x + h) = 1 - (x+h)³ f(x + h) = 1 - (x³ + 3x²h + 3xh² + h³). Using identity Distributing the negative with (x³ + 3x²h + 3xh² + h³), we get f(x + h) = 1 − x³ – 3x²h – 3xh² – h³ b) We know the formula to find the difference quotient is (f(x + h) – f(x)) / h Substituting the values of f(x) = 1 - x³ and f(x + h) = 1 − x³ – 3x²h – 3xh² – h³, we get Difference Quotient = (1 − x³ – 3x²h – 3xh² – h³) – (1 – x³)) / h Distributing the negative with (1 − x³), we get = (1 − x³ – 3x²h – 3xh² – h³ − 1 + x³) / h Simplifying we get, = (−3x²h – 3xh² – h³) / h Factoring h, we get = h(−3x² – 3xh – h²) / h = −3x² − 3xh – h² Question 4 The equation 1.5r + 15 = 2.25r represents the number r of movies you must rent to spend the same amount at each movie store. How many movies must you rent to spend the same amount at each movie store? Answer 1 Let the number of movies rented be r. Now, the equation representing the number of movies r, that must be rented to spend the same amount at each movie store is: 1.5r + 15 = 2.25r 15 = 2.25r - 1.5r 15 = 0.75r 15/0.75 = r 20 = r Answer 2 Step 1: Start with the given equation: 1.5r + 15 = 2.25r. Step 2: We want to isolate the variable r on one side of the equation. To do that, we need to get rid of the constant term 15 on the left side. We can achieve this by subtracting 15 from both sides of the equation: 1.5r + 15 - 15 = 2.25r - 15. Simplifying the equation gives: 1.5r = 2.25r - 15. Step 3: Now, we want to eliminate the coefficient 1.5 in front of r on the left side of the equation. We can do this by dividing both sides of the equation by 1.5: (1.5r / 1.5) = (2.25r - 15) / 1.5 Simplifying further gives: r = (2.25r - 15) / 1.5 Step 4: Next, we simplify the right side of the equation by dividing both terms by 1.5: r = (2.25r/1.5) - (15/1.5) Simplifying gives: r = 1.5r - 10. Step 5: To isolate the variable r on one side of the equation, we need to get rid of the 1.5r term on the right side. We can achieve this by subtracting 1.5r from both sides of the equation: r - 1.5r = 1.5r - 1.5r - 10. Simplifying further gives: -0.5r = -10. Step 6: To solve for r, we divide both sides of the equation by -0.5: (-0.5r/-0.5) = (-10/-0.5) Simplifying gives: r = 20. Therefore, you must rent 20 movies to spend the same amount at each movie store. Answer 3 Answer: 20 Explanation: To solve the equation 1.5r + 15 = 2.25r, representing the number r of movies you must rent to spend the same amount at each movie store, we can follow these steps: Step 1: Subtract 1.5r from both sides of the equation to isolate the variable r: 1.5r + 15 - 1.5r = 2.25r - 1.5r 15 = 2.25r - 1.5r Step 2: Simplify the right side of the equation: 15 = 0.75r Step 3: Divide both sides of the equation by 0.75 to solve for r: 15 / 0.75 = r Step 4: Evaluate the division: 20 = r Therefore, you must rent 20 movies to spend the same amount at each movie store. Answer 4 Move the terms involving the variable r to one side of the equation. Subtract 1.5r from both sides: 1.5r - 1.5r + 15 = 2.25r - 1.5r Simplifying the left side gives: 15 = 0.75r Solve for r by isolating the variable. Divide both sides of the equation by 0.75: 15/0.75 = (0.75r) / 0.75 Simplifying the right side gives: 20 = r Therefore, you must rent 20 movies to spend the same amount at each movie store. Question 5 Describe the shape of a scatter plot that suggests modeling the data with an exponential function. Answer Step 1 The scatter plot that suggests modeling the data with an exponential function Have following properties. The data points on the graph should not be scattered far away from each other. The shape of the curve should either exponentially increasing upwards or exponentially decreasing downwards. Mostly the data points should be above the x - axis. Step 2 The one in which there is a declining trend in the data the graph of such datasets are called Exponentially decaying. While the other in which there is increasing trend is called exponentially increasing.